Question

A railway half-ticket costs half the full ticket. However the reservation charge for all the tickets is constant. One full reserved ticket for a journey is Rupees 525. If the cost of one full and one half reserved ticket for the same journey is Rupees 850, then what is the reservation charge per ticket?

• A. Rupees 120
• B. Rupees 150
• C. Rupees 125
• D. Rupees 115

Hint:

\textbf{Formulating Equations.} When solving word problems, especially those involving costs and charges, it is helpful to define variables for the unknown quantities and then form linear equations based on the information provided in the problem. Solving these equations will give the values of the unknowns.

Answer 1

The Correct Option is C

Let the cost of a full unreserved ticket be \(F\) and the reservation charge per ticket be \(R\). The cost of one full reserved ticket is the sum of the full unreserved ticket cost and the reservation charge: $$ F + R = 525 \quad \cdots (A) $$ The cost of a half unreserved ticket is \( \frac{F}{2} \). The cost of one half reserved ticket is the sum of the half unreserved ticket cost and the reservation charge: $$ \frac{F}{2} + R $$ The cost of one full reserved ticket and one half reserved ticket is given as Rupees 850: $$ (F + R) + \left( \frac{F}{2} + R \right) = 850 $$ $$ F + R + \frac{F}{2} + R = 850 $$ $$ \frac{3F}{2} + 2R = 850 \quad \cdots (2) $$ From equation (A), we have \( F = 525 - R \). Substitute this into equation (2): $$ \frac{3(525 - R)}{2} + 2R = 850 $$ $$ \frac{1575 - 3R}{2} + 2R = 850 $$ Multiply the entire equation by 2 to eliminate the fraction: $$ 1575 - 3R + 4R = 1700 $$ $$ 1575 + R = 1700 $$ $$ R = 1700 - 1575 $$ $$ R = 125 $$ So, the reservation charge per ticket is Rupees 125. We can also find the cost of the full unreserved ticket using equation (A): $$ F + 125 = 525 $$ $$ F = 525 - 125 = 400 $$ The cost of a half unreserved ticket is \( \frac{400}{2} = 200 \). The cost of one full reserved ticket \( = 400 + 125 = 525 \) (as given). The cost of one half reserved ticket \( = 200 + 125 = 325 \). The cost of one full and one half reserved ticket \( = 525 + 325 = 850 \) (as given). Therefore, the reservation charge per ticket is Rupees 125.